To force the mixed strategies to concentrate, we use the imitation game, which has been the driving force in recent advances on complexity of Nash equilibrium (e.g. [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, and in particular the authors' own recent work on communication complexity of Nash equilibrium in potential games [8]}.
Specifically, we add 2 additional players who choose a second point \(y \in [0,1]^2\) , and update the identical utility function to
\(U(x,y) :=-||x-y||_2^2+\epsilon \phi (x),\) 
